Limit of periodic function at infinity

Question

Tried this problem but I am out of ideas.

Let $f: \Bbb R \to \Bbb R$ periodic function with period $T>0$. Prove, using the definition of limits at infinity, that $f$ cannot be infinite while $x \to \infty$ or else $\lim_{x\to\infty}f(x) = \infty$

Answer 1

Suppose that $\lim\limits_{x\rightarrow \infty}f(x)=\infty$, i.e. $\forall E>0~ \exists \Delta(E)$ such that $\forall x$ with $|x|> \Delta$ holds $|f(x)|>E$. For $E>|f(0)|$ and $n\in \mathbb{N}$ such that $nT> \Delta(E)$ we have $|f(nT)|=|f(0)|<E$, which contradicts with our assumption.